Trigonometric Functions

Trigonometric functions, also known as ‘circular functions,’ are the ratio between any two sides of a right triangle: the opposite side, the adjacent side, and the hypotenuse with respect to a reference angle θ. 

There are six trigonometric functions, of which sine, cosine, and tangent functions are basic functions, while secant (sec), cosecant (cosec or csc), and cotangent (cot) are derived from the three basic functions.

In a right-angled triangle ABC, the trigonometric ratios of the six basic trigonometric functions are shown below:

Trigonometric functions have many applications in geometry, trigonometry, and calculus and are commonly used in science and engineering.

Table

Below are the values of the six trigonometric functions for the special angles:

In Quadrants

The sign of each trigonometric function depends on the quadrant in which the angle lies: 

• Quadrant I (0° ≤ θ ≤ 90°): All functions are positive.
• Quadrant II (90° < θ ≤ 180°): sin θ and cosec θ are positive.
• Quadrant III (180° < θ ≤ 270°): tan θ and cot θ are positive.
• Quadrant IV (270° < θ ≤ 360°): cos θ and sec θ are positive.

However, to easily memorize the signs, we use the ASTC rule, which stands for:

• A = All (All functions are positive in Quadrant I)
• S = Sine (Sine and cosecant are positive in Quadrant II)
• T = Tangent (Tangent and cotangent are positive in Quadrant III)
• C = Cosine (Cosine and secant are positive in Quadrant IV)

Graphing

Graphs help visualize the periodic nature of trigonometric functions.

Sine and Cosine (y = sin x, y = cos x)

Amplitude: It is the maximum value of the function. For y = a sin x or y = a cos x, the amplitude is mod a.

Period: It is the length of one complete cycle. For y = cos x, the period is 2π.

Horizontal Shift: It is the displacement of the graph along the x-axis. For y = sin(bx + c), the horizontal (or phase) shift is ${-\dfrac{c}{b}}$​

Vertical Shift: It is the displacement of the graph along the y-axis. For y = sin x + d, the vertical shift is d.

Tangent and Cotangent (y = tan x, y = cot x)

Amplitude: Not defined, as these functions are unbounded.

Period: The period is π.

Asymptotes: These occur where the function is undefined. 

For y = tan x, asymptotes are at ${x=\dfrac{\pi }{2}+n\pi}$

For y = cot x, asymptotes are at x = nπ

Here, n is any integer.

The tangent graph rises and falls infinitely between asymptotes, while the cotangent graph follows a similar pattern with a shift.

Secant and Cosecant (y = sec x, y = cosec x)

The graphs of y = sec x and y = cosec x are U-shaped, reflecting their reciprocal relationships with cos x and sin x, respectively.

Amplitude: Not defined, as these functions are unbounded.

Period: The period is 2π.

Asymptotes: These occur at the zeros of their respective reciprocal functions.

For y = sec x, asymptotes are at ${x=\dfrac{\pi }{2}+n\pi}$

For y = cosec x, asymptotes are at x = nπ

Here, n is any integer.

Domain and Range

The domain and range of trigonometric functions represent the possible input (domain) and output (range) values of each function. Here are the domain and range of the six basic trigonometric functions:

Sine and Cosine (y = sin x, y = cos x)

Domain: The domain is (−∞, ∞), which means both sine and cosine functions are defined for all real numbers.

Range: The range is [−1, 1], which represents the output values are bounded between -1 and 1, inclusive.

Tangent and Cotangent (y = tan x, y = cot x)

Domain: Tangent is undefined where cosine equals zero, while cotangent is undefined where sine equals zero. 

Thus, 

The domain of the tangent function is ${x\neq \dfrac{\pi }{2}+n\pi}$

The domain of the cotangent function is x ≠ nπ

Here, n is any integer.

Range: Both tangent and cotangent functions are unbounded. Thus, the range of the tangent and cotangent functions are (−∞,∞)

Secant and Cosecant (y = sec x, y = cosec x)

Domain: Secant is undefined where cosine equals zero, while cosecant is undefined where sine equals zero.

Thus, 

The domain of the secant function is ${x\neq \dfrac{\pi }{2}+n\pi}$

The domain of the cosecant function is x ≠ nπ

Here, n is any integer.

Range: Both secant and cosecant values are always greater than or equal to 1 or less than or equal to -1.

Thus, the range of the secant and cosecant functions are (-∞, -1] ∪ [1, ∞)

Identities

Below are the identities which are used to solve complex expressions and equations involving trigonometric functions:

Even and Odd Functions

Even functions remain unchanged when the sign of the input angle is reversed.

• cos(-x) = cos x
• sec(-x) = sec x

Odd functions change their sign when the sign of the input angle is reversed.

• sin(-x) = -sin x
• tan(-x) = -tan x
• cot(-x) = -cot x
• cosec(-x) = -cosec x

Periodic Functions

The smallest periodic cycle for sine, cosine, secant, and cosecant is 2π, while for tangent and cotangent, it is π. 

The periodic properties of the trigonometric functions are given below: 

• sin(x + 2nπ) = sin x
• cos(x + 2nπ) = cos x
• tan(x + nπ) = tan x
• cot(x + nπ) = cot x
• cosec(x + 2nπ) = cosec x
• sec(x + 2nπ) = sec x

Here, n is any integer.

Pythagorean Identities

The Pythagorean theorem becomes a Pythagorean identity when expressed in terms of trigonometric functions. There are three such identities:

• sin² x + cos² x = 1
• 1 + tan² x = sec² x
• cosec² x = 1 + cot² x

Sum and Difference Identities

Here are the sum and difference identities for the three basic trigonometric functions:

• sin(x + y) = sin x ⋅ cos y + cos x ⋅ sin y
• sin(x – y) = sin x ⋅ cos y – cos x ⋅ sin y

• cos(x + y) = cos x ⋅ cos y – sin x ⋅ sin y
• cos(x – y) = cos x ⋅ cos y + sin x ⋅ sin y

• tan(x + y) = ${\dfrac{\tan x+\tan y}{1-\tan x\tan y}}$
• tan(x – y) = ${\dfrac{\tan x-\tan y}{1+\tan x\tan y}}$

Product identities

The product-to-sum identities express products of trigonometric functions as sums or differences:

• 2sin x ⋅ cos y = sin(x + y) + sin( x – y)
• 2cos x ⋅ cos y = cos(x + y) + cos(x – y)
• 2sin x ⋅ sin y = cos(x – y) – cos(x + y)

Half-Angle Identities

For θ = ${\dfrac{x}{2}}$, 

• ${\sin \dfrac{x}{2}=\pm \sqrt{\dfrac{1-\cos x}{2}}}$
• ${\cos \dfrac{x}{2}=\pm \sqrt{\dfrac{1+\cos x}{2}}}$
• ${\tan \dfrac{x}{2}=\pm \sqrt{\dfrac{1-\cos x}{1+\cos x}}}$

Double Angle Identities

For θ = 2x,

• sin 2x = 2sin x ⋅ cos x = ${\dfrac{2\tan x}{1-\tan ^{2}x}}$
• cos 2x = cos² x – sin² x = ${\dfrac{1-\tan ^{2}x}{1+\tan ^{2}x}}$
• cos 2x = 2cos² x – 1 = 1 – 2 sin² x
• tan 2x = ${\dfrac{2\tan x}{1+\tan ^{2}x}}$
• sec 2x = ${\dfrac{\sec ^{2}x}{2-\sec ^{2}x}}$
• cosec 2x = ${\dfrac{\sec x\cdot \text{cosec} \ x}{2}}$
• cot 2x = ${\dfrac{\cot ^{2}x-1}{2\cot x}}$

Triple Angle Identities

For θ = 3x,

• sin 3x = 3sin x – 4sin³ x
• cos 3x = 4cos³ x – 3cos x
• tan 3x = ${\dfrac{3\tan x-\tan ^{3}x}{1-3\tan ^{2}x}}$

Sum and Difference of Two Identities

The sum and difference of two trigonometric functions are given by:

• sinx + siny = ${2\sin \left( \dfrac{x+y}{2}\right) \cdot \cos \left( \dfrac{x-y}{2}\right)}$
• sinx – siny = ${2\cos \left( \dfrac{x+y}{2}\right) \cdot \sin \left( \dfrac{x-y}{2}\right)}$
• cosx + cosy = ${2\cos \left( \dfrac{x+y}{2}\right) \cdot \cos \left( \dfrac{x-y}{2}\right)}$
• cosx – cosy = ${-2\sin \left( \dfrac{x+y}{2}\right) \cdot \sin \left( \dfrac{x-y}{2}\right)}$

Derivatives

Differentiating trigonometric functions helps determine the slope of the tangent line to their curve at any given point.

For example, 

The derivative of sinx is cos x. By applying the value of x (in degrees or radians) to cos x, the slope of the tangent to the sinx curve at that point can be determined. 

Here are the formulas for the derivatives of the trigonometric functions:

• ${\dfrac{d}{dx}\left( \sin x\right) =\cos x}$
• ${\dfrac{d}{dx}\left( \cos x\right) =-\sin x}$
• ${\dfrac{d}{dx}\left( \tan x\right) =\sec ^{2}x}$
• ${\dfrac{d}{dx}\left( \cot x\right) =-\text{cosec} ^{2} \  x}$
• ${\dfrac{d}{dx}\left( \sec x\right) =\sec x\cdot \tan x}$
• ${\dfrac{d}{dx}\left( \text{cosec} \  x\right) =-\text{cosec} \  x\cdot \cot x}$

Integration

The integration of trigonometric functions provides the area under the curve of the function. In simpler terms, it reverses the process of differentiation. 

Here are the formulas for the integrations of the trigonometric functions:

• ${\int \sin xdx=-\cos x+C}$
• ${\int \cos xdx=\sin x+C}$
• ${\int \sec ^{2}xdx=\tan x+C}$
• ${\int \text{cosec} ^{2} \  xdx=-\cot x+C}$
• ${\int \sec x\cdot \tan xdx=\sec x+C}$
• ${\int \text{cosec} \  x\cdot \cot xdx=-\text{cosec} \  x+C}$
• ${\int \tan xdx=\ln \left| \sec x\right| +C}$
• ${\int \cot xdx=\ln \left| \sin x\right| +C}$
• ${\int \sec xdx=\ln \left| \sec x+\tan x\right| +C}$
• ${\int \text{cosec} \  xdx=-\ln \left| \text{cosec} \  x+\cot x\right| +C}$

Here, C is an integral constant.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of basic trigonometric functions. They are used to find the angle corresponding to a given trigonometric ratio.

For example, if sin y = x is known, then the angle y can be obtained by y = sin^-1 x 

Here is the list of inverse trigonometric functions:

1. Arcsine (sin^-1 or arcsin): y = sin^-1 x ⇒ sin y = x
2. Arccosine (cos^-1 or arccos): y = cos^-1 x ⇒ cos y = x
3. Arctangent (tan^-1 or arctan): y = tan^-1 x ⇒ tan y = x
4. Arcsecant (sec^-1 or arcsec): y = sec^-1 x ⇒ sec y = x
5. Arccosecant (cosec^-1 or arccosec): y = cosec^-1 x ⇒ cosec y = x
6. Arccotangent (cot^-1 or arccot): y = cot^-1 x ⇒ cot y = x

Solved Examples